GCSE · Functions

Composite functionsExam-style questions

Why fg(x) means "g first, then f", how to chain outputs into inputs for numbers and for algebra, and why the order almost always matters.

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16 questions · GCSE (Higher) level · mark scheme and worked solution on each · no sign-up to try

DifficultyAllLevel 3Level 4Level 5
  1. 1
    f(x)=2x+3g(x)=x2−1\mathrm{f}(x) = 2x + 3 \qquad \mathrm{g}(x) = x^2 - 1

    (a) Show that gf(x)=4x2+12x+8\mathrm{gf}(x) = 4x^2 + 12x + 8 (2 marks)

    (b) Solve gf(x)=8\mathrm{gf}(x) = 8 (2 marks)
    Level 4 · 4 marksTry it
  2. 2
    f(x)=3x−1g(x)=x2+1\mathrm{f}(x) = 3x - 1 \qquad \mathrm{g}(x) = x^2 + 1

    (a) Work out fg(2)\mathrm{fg}(2). [2]

    (b) Solve fg(x)=gf(x)\mathrm{fg}(x) = \mathrm{gf}(x). [3]
    Level 4 · 5 marksTry it
  3. 3
    The functions f and g are given by

    f(x)=18x−2andg(x)=7−4x\mathrm{f}(x) = \frac{18}{x - 2} \qquad \text{and} \qquad \mathrm{g}(x) = 7 - 4x
    (a) Find f(−4)\mathrm{f}(-4)

    (1 mark)
    (b) Find fg(−1)\mathrm{fg}(-1)

    (2 marks)
    (c) Find g−1(15)\mathrm{g}^{-1}(15)

    (2 marks)
    Level 4 · 5 marksTry it
  4. 4
    f(x)=2x+1f(x) = 2x + 1 and g(x)=x2−3g(x) = x^2 - 3

    (a) Work out fg(3)fg(3)

    (b) Work out gf(3)gf(3)
    Level 4 · 4 marksTry it
  5. 5
    f(x)=x2f(x) = x^2 and g(x)=2x−1g(x) = 2x - 1

    (a) Find fg(x)fg(x)

    (b) Show that fg(2)≠gf(2)fg(2) \neq gf(2)
    Level 4 · 4 marksTry it

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